Found problems: 916
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$
(Walther Janous)
Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.
Prove that the equation
$$1320x^3=(y_1+y_2+y_3+y_4)(z_1+z_2+z_3+z_4)(t_1+t_2+t_3+t_4+t_5)$$
has infinitely many solutions in the set of Fibonacci numbers.
[i]Proposed by Mihály Bencze[/i]
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to
$(m - n)^2 = kmn + m + n$.
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:
\[x^2 - (n^2 +1)y^2 = n^2.\]
(25 points)
Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]
Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers.
In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.
Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$
Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$.
(a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$.
(b) Find an example of eleven consecutive squares whose sum is a square.
Can anyone help me with this?
Thanks.
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$,
which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square .
Remark: $(7, 4) \ne (4, 7)$
Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.
Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$
Solve the following system of equations in integer numbers:
$$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$.
(W. Janous, Innsbruck)
Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.
Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that
$(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$.
(Patrik Bak, Slovakia)
Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.